Hooper's paradox is a

falsidical paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

based on an optical illusion. A geometric shape with an area of 32 units is dissected into four parts, which afterwards get assembled into a rectangle with an area of only 30 units.

Explanation

Upon close inspection one can notice that the triangles of the dissected shape are not identical to the triangles in the rectangle. The length of the shorter side at the right angle measures 2 units in the original shape but only 1.8 units in the rectangle. This means, the real triangles of the original shape overlap in the rectangle. The overlapping area is a parallelogram, the diagonals and sides of which can be computed via the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...

. :

d_1=\sqrt=\sqrt

d_2=\sqrt=\sqrt

s_1=\sqrt=\sqrt

s_2=\sqrt=\sqrt

The area of this parallelogram can determined using

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-centur ...

for triangles. This yields :

s=\frac=\frac

for the halved circumference of the triangle (half of the parallelogram) and with that for the area of the parallelogram :

&=2 \end

. So the overlapping area of the two triangles accounts exactly for the vanished area of 2 units.

History

William Hooper published the paradox 1774 in his book ''Rational Recreations'' in which he named it ''The geometric money''. The 1774 edition of his book still contained a false drawing, which got corrected in the 1782 edition. However Hooper was not the first to publish this geometric fallacy, since Hooper's book was largely an adaption of

Edmé-Gilles Guyot Edmé-Gilles Guyot (1706–1786) was a French mail clerk, physician, postmaster, cartographer, inventor and author on the subject of mathematics, physics and magic. He experimented with optical illusions and with the theory behind performance magic. ...

's ''Nouvelles récréations physiques et mathétiques'', which had been published in France in 1769. The description in this book contains the same false drawing as in Hooper's book, but it got corrected in a later edition as well.

References

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of L ...

: ''Mathematics, Magic and Mystery''. Courier (Dover), 1956, , S. 129–155 *Greg N. Frederickson: ''Dissections: Plane and Fancy''. Cambridge University Press, 2003, , chapter 23, pp. 268–277 in particular pp. 271–274
online update to chapter 23
* Simon During: ''Modern Enchantments: The Cultural Power of Secular Magic''. Harvard University Press, 2004, {{ISBN, 978-0674013711, p. 87 * William Hooper: ''Rational Recreations''. London, 1774, pp
286–287
(faulty 1st edition) * William Hooper: ''Rational Recreations''. London, 1782, pp
286–287
(corrected 2nd edition)

External links

''Hooper's Paradox: How Is It Possible?''
auf cut-the-knot.org * Mariano Tomatis
''Curse of the crystal skulls and other vanishing area puzzles''''Rational recreations'' by W. Hooper
Optical illusions Elementary mathematics Mathematical paradoxes Recreational mathematics Logic puzzles Geometric dissection

Explanation

History

See also

References

External links